Lesson Objectives
• Demonstrate an understanding of the distance formula
• Demonstrate an understanding of the midpoint formula
• Learn how to find the distance between two complex numbers
• Learn how to use the midpoint formula with complex numbers

## How to Find the Distance Between Two Complex Numbers

At this point, we have been working with the complex plane for quite a while. We've seen how to plot complex numbers on the complex plane and also learned about how to find the absolute value of a complex number. Now, let's go a little further and think about how we can find the distance between two complex numbers on the complex plane.
We already know how to use the distance formula to find the distance between two points on the coordinate plane.

### Distance Formula

$$d=\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ When we work with complex numbers, the formula is pretty much the same. The difference is we think about the vertical change as the difference along the imaginary axis and we think about the horizontal change as the difference along the real axis. Since we think about a complex number as: a + bi, where a is the real part and b is the imaginary part, we can revise our formula.
The distance between two complex numbers is given as: $$d=\sqrt{(a_2 - a_1)^2 + (b_2 - b_1)^2}$$ Example #1: Find the distance between the given complex numbers. $$5 + 3i$$ $$-1 - 7i$$ Let's label a1, a2, b1, and b2. It doesn't matter which is chosen as the first or second complex number, just be consistent. $$a_2=-1$$ $$a_1=5$$ $$b_2=-7$$ $$b_1=3$$ Now, we can plug into our formula: $$d=\sqrt{(a_2 - a_1)^2 + (b_2 - b_1)^2}$$ $$d=\sqrt{(-1 - 5)^2 + (-7 - 3)^2}$$ $$d=\sqrt{(-6)^2 + (-10)^2}$$ $$d=\sqrt{36 + 100}$$ $$d=\sqrt{136}$$ $$d=2\sqrt{34}$$ Example #2: Find the distance between the given complex numbers. $$12 - 5i$$ $$13 + i$$ Let's label a1, a2, b1, and b2. It doesn't matter which is chosen as the first or second complex number, just be consistent. $$a_2=13$$ $$a_1=12$$ $$b_2=1$$ $$b_1=-5$$ Now, we can plug into our formula: $$d=\sqrt{(a_2 - a_1)^2 + (b_2 - b_1)^2}$$ $$d=\sqrt{(13 - 12)^2 + (1 - (-5))^2}$$ $$d=\sqrt{(1)^2 + (6)^2}$$ $$d=\sqrt{37}$$

## Midpoint Formula with Complex Numbers

Let's move on and think about the midpoint formula in the complex plane. We previously saw that the midpoint formula can be used to find the coordinates of the midpoint of a line segment. In the real number system, if two given points (x1, y1) and (x2, y2) are the endpoints of a line segment, then the coordinates of the midpoint are given as: $$M=\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ We can see the x-coordinate of the midpoint of the line segment is just the average of the x-coordinates of the segment’s endpoints. Similarly, we can see the y-coordinate of the midpoint of the line segment is just the average of the y-coordinates of the segment’s endpoints.
If we have a line segment with two endpoints (a1,b1) and (a2,b2) in the complex plane. We can adjust our formula: $$M=\left(\frac{a_1 + a_2}{2}, \frac{b_1 + b_2}{2}\right)$$ Note: A complex number is of the form: a + bi
These coordinates will give us the real location and the imaginary location on the complex plane. We can use this information to state the midpoint as a complex number. $$M=\frac{a_1 + a_2}{2}+ \frac{b_1 + b_2}{2}i$$ Example #3: Find the midpoint of the line segment zw. $$z=3 + 9i$$ $$w=5 + 17i$$ We can pick either z or w to be the first or second complex number, just be consistent. $$a_1=3$$ $$b_1=9$$ $$a_2=5$$ $$b_2=17$$ Once you have labeled everything, just plug into the formula. $$M=\left(\frac{a_1 + a_2}{2}, \frac{b_1 + b_2}{2}\right)$$ $$M=\left(\frac{3 + 5}{2}, \frac{9 + 17}{2}\right)$$ $$M=\left(\frac{8}{2}, \frac{26}{2}\right)$$ $$M=\left(4, 13\right)$$ So 4 is our a value or real part for the complex number, and 13 is our b value or imaginary part for the complex number. To write our midpoint as a complex number: $$4 + 13i$$

#### Skills Check:

Example #1

Find the distance between z and w. $$z=2 - 10i$$ $$w=5 + i$$

A
$$\sqrt{21}$$
B
$$\sqrt{17}$$
C
$$2\sqrt{17}$$
D
$$2\sqrt{130}$$
E
$$\sqrt{130}$$

Example #2

Find the distance between z and w. $$z=-3 + 2i$$ $$w=4i$$

A
$$\sqrt{17}$$
B
$$2\sqrt{5}$$
C
$$\sqrt{13}$$
D
$$\sqrt{2}$$
E
$$\sqrt{19}$$

Example #3

Find the midpoint of zw. $$z=6 + 9i$$ $$w=-2 + 11i$$

A
$$-4 - 9i$$
B
$$-10i$$
C
$$3 + i$$
D
$$2 + 10i$$
E
$$-1 + 4i$$         